After alerting my neighbour that tree roots were lifting my walk path, neighbour amicably cut tree down and applied stump killer. However tree roots (suckers) have appeared in my lawn and travelled roots have lifted slabs of a distant patio. My now angry neighbour believes I should deal with the roots in my garden and is quite adamant that nothing needs to be done other than wait for the tree roots to die down having cut the tree down.

I can't walk up my path without tripping, and I am afraid of baby trees taking over my small garden. Am I right in thinking that I will be unsuccessful with a small claims action to recover the costs for any remedial work I undertake to rectify my walk path or remove invasive roots in my garden.

because if a contract existed you would have grounds for a claim.
otherwise what would your claim be based on - that he "ought" to pay for the work because it was his tree? it doesn't work like that because he didn't decide where the tree grew - Mother Nature did.